Symplectic representation

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In mathematics and theoretical physics, a pseudoreal representation is a group representation that is equivalent to its complex conjugate, but that is not a real representation. That is, it satisfies the obvious necessary condition to be equivalent to a representation by means of real matrices; but is not actually equivalent to such a representation. Such representations can exist essentially for the same reasons that the quaternions, a division algebra over the real numbers, can exist. A symplectic representation is a particular kind of pseudoreal representation, of a finite group or more generally of a compact group, that is an essentially quaternionic representation. Such representations, amongst irreducible representations, can be picked out by the Frobenius-Schur indicator.

When it comes to representations that are not irreducible, one could give an example of a direct sum of a real representation and a symplectic representation, as satisfying the stated condition to be pseudoreal; this however is less useful.

An irreducible group representation that is neither real nor symplectic is a complex representation.

Approaching the question from the direction of Schur's lemma, i.e. module endomorphisms for the representation space, there exists for a symplectic representation an antilinear map

$j:V\to V$

that commutes with the elements of the group, but it satisfies

j 2 = - 1

Pseudoreal representations are often called quaternionic representations because the group elements can be expressed as matrices whose entries are quaternions. Another way to look at this is to look at the real group algebra, and identify it as a direct sum of simple R-algebras. These will be (Artin-Wedderburn theorem) matrix algebras over the real numbers or the quaternions. The latter case is responsible for the phenomenon of pseudoreal representation.

Examples of pseudoreal representations are the spinors of SO(d) group where d equals $3 + 8 k,4 + 8 k$ , and $5 + 8 k$ dimensions, where k is an integer. In physics, one often encounters the spinors of SO(d,1). For them, the reality agrees with the spinors of SO(d-1).

As for the real form of the simple Lie groups, pseudoreal representations only exist for the Lie groups of type $A_{4k+1},\ B_{4k+1},\ B_{4k+2},\ C_k,\ D_{4k+2}$ and for E₇.